Track to bone constraints lag behind animation/don't update correctly
Idk if I got your setup, but imho you can do something like that with 2 armatures:

A first one-bone armature, and an empty parented to it (tip of the bone)



then another 2-bones armature, where each bone has different constraints: the bottom bone copies rotation of first armature bone



and the horizontal bone tracks the empty



to show it working, I parented 3 cylinders each to its bone, and key framed rotation of first armature only

in viewport:



and rendered:



gif timing seems weird but I rendered to h264 avi and it works really well, no lagging

here the .blend